Probability Density

Editor’s Note: In this section, I’ll break down some of the key aspects in probability theory that shape the basis for this website. Next, I look how probability density plays a crucial role in sports statistical analysis. Please note these explanations won’t be 100% up to mathematical textbook standards, simply because these explanations need to be shaped into a sports context. If there are any concerns or criticisms about the process of applying probability theory into a sports context, please contact me at tabmathletics@gmail.com.

Once there’s good reason to believe there’s normal distribution in a sample set, whether the sample size is sufficiently large or the necessary tests pass, it’s important to note how probability density explains the likelihood of certain numbers being achieved among established intervals in each set.

With normal distribution present, the density decreases as a specific sample moves farther away from the mean (which is the central point). That means the numbers that are extremely far from the mean are highly unlikely to happen. Under these conditions, we can safely conclude that certain rare statistical feats will not repeat themselves. Without getting into specific percentages, simply because we’d have to track to the beginning of a league’s stat keeping to established the true sample size, some of these stats have about one percent odds of repeating. Therefore, it’s most logical to conclude that something in the “99 percent” will happen.

This all can be easily described with a simple graph showing probability density in normal distribution:

Note that the three areas above total 99.73 percent. Therefore, finding something outside those ranges would involve an event that happens less than once every 100 times. Obviously, it will be tough to find something three standard deviations away (that omega symbol means one standard deviation) from the mean, but we may be able to find something that’s near the 97 or 99 percentile on the outer tails of a graph much like this one.

My job in this website will be to find out which teams and players put up numbers on the outer tails of certain statistical graphs. Just by playing the odds, we can make safe conclusions about the said teams and players regressing to the mean, something that will be discussed in the final chapter of this section.